Sauté Aromatics: In a large pot over medium heat, melt the butter with the olive oil. Add the diced onion and sauté until translucent, about 5 minutes.
Add Jalapeños and Garlic: Stir in the chopped jalapeños and cook for another 3 minutes until they begin to soften. Add the minced garlic and cook for an additional minute until fragrant.
Incorporate Spices: Sprinkle in the ground cumin, smoked paprika, salt, and black pepper. Stir well to evenly coat the vegetables with the spices.
Create Roux: Add the all-purpose flour to the pot, stirring continuously for about 2 minutes to cook out the raw flour taste.
Add Broth: Gradually pour in the chicken or vegetable broth, stirring constantly to prevent lumps. Bring the mixture to a boil, then reduce the heat and let it simmer for 10 minutes.
Blend Soup: Using an immersion blender, purée the soup until smooth. If using a countertop blender, allow the soup to cool slightly before blending in batches, then return it to the pot.
Add Cream and Cheese: Stir in the heavy cream and shredded cheese, mixing until the cheese has fully melted and the soup is creamy.
Finalize Seasoning: Add the lime juice and adjust seasoning with additional salt and pepper if needed.
Serve: Ladle the soup into bowls and garnish with freshly chopped cilantro. Serve hot.